162 CHAPTER 4. UNDERSTANDING FORWARD EXCHANGE RATES FOR CURRENCY
the replications for the other transactions and check that they generate the gross
returns shown in the rightmost column.
replicand replication output value#
t-to-T 2 deposit spot depositttoT 1 , rolled over
forwardT 1 toT 2(1 +rt,T 1 )(1 +rft,T 1 ,T 2 )forward depositT 1 toT 2 1+1+rrt,Tt,T^21spot depositttoT (^1) 1+1+rfrt,T^2
t,T 1 ,T 2
t-to-T 2 loan (1+rt,T^1
1 )(1+rft,T 1 ,T 2 )
forward loanT 1 toT 2 1+1+rrt,Tt,T^12
spot loanttoT 1
1+rft,T 1 ,T 2
1+rt,T 2
#: output is computed for input value equal to unity.
.
But the diagram shows not just the replication possibilities: there are also two
no-arb constraints inside each money market, corresponding to, respectively, the
clockwise and counterclockwise roundtrips. You start for instance in the time-T 2
box, issuing apnnote datedT 2. You discount it immediately and make a synthetic
depositttoT 2. The constraint is that the proceeds of this deposit be no higher
than 1, the amount you owe to the holder of thepn:
HCT 2 →HCt→HCT 1 →HCT 2
1 ×1
1 +rt,T 2
×(1 +rt,T 1 )×(1 +rft,T 1 ,T 2 )≤ 1. (4.49)DoItYourself problem 4.10
Identify the other no-arb trip in the home money market and write the corresponding
constraint. Combine it with constraint [4.49] and check that you find back Equation
[4.47].
To seasoned arbitrageurs like you it is easy to add bid- and ask-superscripts to
the rates of return (and to the exchange rates, while you are at it). The no-arb
constraints still are [synthetic bid]≤[ask] and [bid] ≤[synthetic ask], with the
synthetics computed from the worst-possible-combination versions of the perfect-
market replication that you just worked out yourself.
Before we move to other markets, there is another set of no-arb constraints and
shopping-around opportunities to be discussed, namely those created via interna-
tional linkages rather than relations within each money market. Remember one can